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August  2015, 35(8): 3707-3719. doi: 10.3934/dcds.2015.35.3707

Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

Mitsuru Shibayama 1,

1. 

Division of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

Received  April 2014 Revised  December 2014 Published  February 2015

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be taken from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.
Keywords: blowing-up technique, homogeneous Hamiltonian systems., Integrability.
Mathematics Subject Classification: Primary: 37J30; Secondary: 70F1.
Citation: Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics,, $2^{nd}$ edition, (1989).  doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

H. Bruns, Über die Integrale des Vielkörper-Problems,, Acta Math., 11 (1887), 25.  doi: 10.1007/BF02612319.  Google Scholar

[3]

R. L. Devaney, Triple collision in the planar isosceles three-body problem,, Invent. Math., 60 (1980), 249.  doi: 10.1007/BF01390017.  Google Scholar

[4]

R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem,, Celestial Mech., 28 (1982), 25.  doi: 10.1007/BF01230657.  Google Scholar

[5]

G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations,, Discrete Contin. Dyn. Syst., 34 (2014), 4589.  doi: 10.3934/dcds.2014.34.4589.  Google Scholar

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S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177.  doi: 10.1007/BF02592182.  Google Scholar

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R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.  doi: 10.1007/BF01390175.  Google Scholar

[8]

R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem,, SIAM J. Math. Anal., 15 (1984), 857.  doi: 10.1137/0515065.  Google Scholar

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J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems,, Birkhaeuser Basel, (1999).  doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[10]

J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113.   Google Scholar

[11]

H. Poincaré, New Methods of Celestial Mechanics Vol. 1,, American Institute of Physics, (1993).   Google Scholar

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M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates,, J. Phys. A: Math. Gen., 30 (1997), 5869.  doi: 10.1088/0305-4470/30/16/026.  Google Scholar

[13]

M. Shibayama, Non-integrability of the collinear three-body problem,, Discrete Contin. Dyn. Syst., 30 (2011), 299.  doi: 10.3934/dcds.2011.30.299.  Google Scholar

[14]

M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem,, Nonlinearity, 22 (2009), 2377.  doi: 10.1088/0951-7715/22/10/004.  Google Scholar

[15]

H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems,, Physica, 21 (1986), 163.  doi: 10.1016/0167-2789(86)90087-4.  Google Scholar

[16]

H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential,, Physica, 29 (1987), 128.  doi: 10.1016/0167-2789(87)90050-9.  Google Scholar

[17]

M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians,, RIMS Kokyuroku Bessatsu, B40 (2013), 177.   Google Scholar

[18]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I., Funktsional. Anal. i Prilozhen., 16 (1982), 30.   Google Scholar

[19]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II., Funktsional. Anal. i Prilozhen., 17 (1983), 8.   Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics,, $2^{nd}$ edition, (1989).  doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

H. Bruns, Über die Integrale des Vielkörper-Problems,, Acta Math., 11 (1887), 25.  doi: 10.1007/BF02612319.  Google Scholar

[3]

R. L. Devaney, Triple collision in the planar isosceles three-body problem,, Invent. Math., 60 (1980), 249.  doi: 10.1007/BF01390017.  Google Scholar

[4]

R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem,, Celestial Mech., 28 (1982), 25.  doi: 10.1007/BF01230657.  Google Scholar

[5]

G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations,, Discrete Contin. Dyn. Syst., 34 (2014), 4589.  doi: 10.3934/dcds.2014.34.4589.  Google Scholar

[6]

S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177.  doi: 10.1007/BF02592182.  Google Scholar

[7]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.  doi: 10.1007/BF01390175.  Google Scholar

[8]

R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem,, SIAM J. Math. Anal., 15 (1984), 857.  doi: 10.1137/0515065.  Google Scholar

[9]

J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems,, Birkhaeuser Basel, (1999).  doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[10]

J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113.   Google Scholar

[11]

H. Poincaré, New Methods of Celestial Mechanics Vol. 1,, American Institute of Physics, (1993).   Google Scholar

[12]

M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates,, J. Phys. A: Math. Gen., 30 (1997), 5869.  doi: 10.1088/0305-4470/30/16/026.  Google Scholar

[13]

M. Shibayama, Non-integrability of the collinear three-body problem,, Discrete Contin. Dyn. Syst., 30 (2011), 299.  doi: 10.3934/dcds.2011.30.299.  Google Scholar

[14]

M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem,, Nonlinearity, 22 (2009), 2377.  doi: 10.1088/0951-7715/22/10/004.  Google Scholar

[15]

H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems,, Physica, 21 (1986), 163.  doi: 10.1016/0167-2789(86)90087-4.  Google Scholar

[16]

H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential,, Physica, 29 (1987), 128.  doi: 10.1016/0167-2789(87)90050-9.  Google Scholar

[17]

M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians,, RIMS Kokyuroku Bessatsu, B40 (2013), 177.   Google Scholar

[18]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I., Funktsional. Anal. i Prilozhen., 16 (1982), 30.   Google Scholar

[19]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II., Funktsional. Anal. i Prilozhen., 17 (1983), 8.   Google Scholar

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